Difference polynomials

In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.

Definition

The general difference polynomial sequence is given by

p_{n}(z)={\frac {z}{n}}{{z-\beta n-1} \choose {n-1}}

where ${z \choose n}$ the binomial coefficient. For $\beta =0$ , the generated polynomials $p_{n}(z)$ are the Newton polynomials

p_{n}(z)={z \choose n}={\frac {z(z-1)\cdots (z-n+1)}{n!}}

The case of $\beta =1$ generates Selberg's polynomials, and the case of $\beta =-1/2$ generates Stirling's interpolation polynomials.

The generating function for the general difference polynomials is given by

e^{zt}=\sum _{n=0}^{\infty }p_{n}(z)\left[\left(e^{t}-1\right)e^{\beta t}\right]^{n}

This generating function can be brought into the form of the generalized Appell representation

K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}

by setting $A(w)=1$ , $\Psi (x)=e^{x}$ , $g(w)=t$ and $w=(e^{t}-1)e^{\beta t}$ .

The

The polynomial sequences are in general summable only for analytic functions of less than exponential type.

Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.